Renormalization and Twists

When I try to get a deep understanding of what E=hv quantization means, I see that for a given frequency, the energy of the smallest particle possible is quantized.  For such a particle to emerge from a field, I have to find out a logical or geometric way for the field to be quantized, and twists within a unitary background vector field appear to be the most obvious case.  This implies a force that exerts a tendency to tie down (for both ends of the twist region) to the background state direction, and one such “force” is the specification of continuity.  Unfortunately, I have yet to find a twist solution that doesn’t have a discontinuity somewhere, a contradiction in the theory’s assumptions.

I am going to specify the field as a spherical coordinate pair of angles, say f = (a,b) since this is a unitary field.  I can create a mathematical equation that specifies that curl in space equals displacement in time, and specify boundary conditions f(0) = (0,0), f(1) = (0,Pi), f(2) = (0,2 Pi) and see if I can show there are no continuous solutions.  This may be simple to do mathematically, I don’t know yet–the field equivalance of angles 0 and 2 Pi require some interesting complications to a proof.  I think I can show geometrically that there cannot be a solution, but the differential equation method should be conclusive.

Yet, somehow, I feel there has to be some kind of a solution.  I was thinking about renormalization and perturbation theory, how this alpha value probability is summed over all possible interactions.  There are first order terms (direct field photon interactions) and second order terms (virtual electron/positron) pair productions appearing in the first order interactions, third order terms as those interactions get pair productions of their own, and so on.   Summing over all these interactions iteratively gets us closer and closer to the right answer.  This is thought of as an electron interacting with its own field, which is a recursive problem since these interactions are mediated (via photons) with more electrons.  I just keep thinking that this recursion is analogous to a Taylor series expansion of an analytic function, we have the Taylor series but we don’t have the mathematical vocabulary/toolset to convert this series to an analytic function.  I have no doubt that every smart physics PhD candidate in the last 70 years has tried to find the analytic solution, and so far as I can see, no one has succeeded.  So I’m going to succeed?  I have no illusions on that matter, of course not.

But I’ve convinced that the two concepts I have mentioned above (E=hv quantization twists, renormalization perturbation) have to meet somehow.  Either these two concepts have to merge in the middle, or twists are the wrong way to express E=hv quantization.  I just can’t bring myself to throw out twists–it is such a clear-cut way to express a continuous modulo function on a field.  Perhaps there’s some kind of compromise where the field magnitude is not unitary–but if you do that, you cannot prevent dispersement of the field entity (that’s why Maxwell’s equations cannot yield a solution to quantization).  By adding the two assumptions (unitary field, localized background direction),  the necessary and sufficient conditions for quantized particles emerging from a field then exist.  The trouble is, just because it is necessary and sufficient doesn’t mean it is right…



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